For a cyclic quadrilateral, the product of the lengths of the diagonals equals the sum of the products of the two pairs of opposite sides. Attributed to Ptolemy, it generalizes the Pythagorean theorem and gives a route to several classical trigonometric identities.
Facts
StatementFor a cyclic quadrilateral with vertices A, B, C and D in order, the product of the two diagonals equals the sum of the products of the two pairs of opposite sides: AC times BD equals AB times CD plus BC times AD. 2 Proof YearApproximate: Ptolemy's theorem appears in his Almagest, whose completion is dated to about 150 on the basis of the Canopus inscription of 147 or 148, a quarter century after Ptolemy began observing. Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Sources
1. Wikipedia: Ptolemy's theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
If the vertices of the cyclic quadrilateral are A, B, C, and D in order, then the theorem states that: A C ⋅ B D = A B ⋅ C D + B C ⋅ A D This relation may be verbally expressed as follows: If a quadrilateral is cyclic then the product of the lengths of its diagonals is equal to the sum of the products of the lengths of the pairs of opposite sides.
2. Ptolemy's Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, theorem statement sentenceQuote, lead paragraph, theorem statement sentence
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle).
3. Almagest (Wikipedia)
Wikimedia Foundationdating of composition, Canopic Inscription discussionQuote, dating of composition, Canopic Inscription discussion
Ptolemy set up a public inscription at Canopus, Egypt, in 147 or 148. Norman T. Hamilton found that the version of Ptolemy's models set out in the Canopic Inscription was earlier than the version in the Almagest. Hence the Almagest could not have been completed before about 150, a quarter-century after Ptolemy began observing.
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